Goodness of Fit of Frequency Distributions. 375 



Further, 



2(^-1) = 34, ?(?-2)/N = -32. 



Accordingly, 



%-!)+£ + 



9 , Q(q—2) 



= 35-106,272,53, 



and 



H ' N 

 a y2 = {35*106,272,53}* = 5*9256. 



Since % 2 is only 10 '27, this indicates that the material 

 might quite easily be a sample from the Gaussian given. 

 Now 



F q ( x 2 ) = -8909 and P ? _ 2 ( % 2 ) = '8016, 

 thus 



MP«(x ! )-p 1 - s (x ! )( 



This leads to 



ilW)-P ? -2Ml --04 



cr p = 5-9256 x '04465 = '2646, 



and probable error of P = *1785. 



The result may therefore be given as 



P =-89 + -18. 



It will be seen that the test of goodness of fit is subject to 

 a larger probable error than has hitherto been realized. It 

 will not in this case affect our judgment, for P is so 

 large that the large probable error will not reduce it to a 

 really small probability, but it indicates that, especially in 

 considering relative goodness of fit, we must proceed with 

 caution in this matter. 



Illustration II. — I will consider the application of the 

 present theory to fourfold tables, and take as examples two 

 tables cited on p. xxxiv of my 'Tables for Statisticians' for 

 characters of fruit and flower in .Datura and for Eye-colour 

 in father and son. The tables are : — 



(i.) Bah 



(ii.) Eye-colour in Father and Son. 



fe 





Colour of Flower. 



Violet. White. 



Totals. 



68 

 15 



Prickly ... 

 Smooth ... 



47 1 21 

 12 3 



Totals.. 



59 24 



83 





Father. 



Light. 



Not- ; r , , 



Light. rotai8 ' 



Light 



Not-Light. 



471 

 151 



148 619 

 230 1 381 



Totals... 



622 378 j 1000 



1 



